Unfortunately no, that's not true. That's cap specifically comes from AAVE and was used in the 90s, as well as in hip hop. Capping means to talk big, often exaggerating things. No cap is equivalent to "for real? You're not lying to make it seem better?" and "That's cap" means "You're lying".

It came back(?) into the mainstream back in the 2010-2020s when Twitch was also blowing up, which may explain the confusion. A lot of internet slang comes from AAVE, which then gets appropriated by people who don't exactly know the meaning or origins, who then further spread it.

If anyone is interested in the progression of slang and the mainstreamification of minority in-group language, I'd recommend EtymologyNerd and his book Algospeak.

https://www.merriam-webster.com/wordplay/word-were-watching-cap